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Theorem difeq12 3289
Description: Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
difeq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )

Proof of Theorem difeq12
StepHypRef Expression
1 difeq1 3287 . 2  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
2 difeq2 3288 . 2  |-  ( C  =  D  ->  ( B  \  C )  =  ( B  \  D
) )
31, 2sylan9eq 2335 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C
)  =  ( B 
\  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    \ cdif 3149
This theorem is referenced by:  resdif  5494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-dif 3155
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